Problem 1 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(A\) and \(B\) are symmetric matrices and \(A B=B A\), then \(A^{-1} B\) is a (A) symmetric matrix (B) skew-symmetric matrix (C) identity matrix (D) None of these

Step-by-Step Solution

Answer

(D) None of these

Step 1: Review Properties of Symmetric Matrices

Recall that a symmetric matrix is one that is equal to its transpose, i.e., if \( A \) is symmetric, then \( A = A^T \). Similarly, for \( B \), if \( B \) is symmetric, then \( B = B^T \).

Step 2: Understand Given Conditions

We are given that \( A \) and \( B \) are symmetric matrices, and they commute with each other, i.e., \( A B = B A \).

Step 3: Explore Symmetry in Matrix Product

For the product of two commuting symmetric matrices, \( (A B)^T = B^T A^T = B A = A B \). Hence, \( A B \) is symmetric.

Step 4: Consider Inverse of Matrix

If \( A \) is invertible, then \( A^{-1} \) exists, and we investigate if \( A^{-1} B \) maintains symmetry.

Step 5: Compute Transpose of \(A^{-1} B\)

Since \( (A^{-1}B)^T = B^T (A^{-1})^T = B A^{-1} \) (since \( B \) is symmetric, so \( B^T = B \)). We need \( A^{-1}B = B A^{-1} \) for \( A^{-1}B \) to be symmetric.

Step 6: Check Commutativity of \(A^{-1} B \)

As \( A^{-1} B \) is not ensured to commute with \( A \), it does not maintain general symmetry or specific commutativity unless additional information is given about \( A \) or \( B \).

Step 7: Conclusion: Identify Matrix Type

Without further conditions to ensure symmetry or skew-symmetry, \( A^{-1} B \) does not fall neatly into any specific category of symmetry given only the initial conditions provided.

Key Concepts

Matrix Commutativity

When discussing matrices, commutativity is crucial in understanding how matrices interact. In mathematics, two matrices are said to commute if their product is the same regardless of the order in which they are multiplied. This means that if matrices \( A \) and \( B \) commute, then \( A B = B A \). Commutativity is a special property because, in general, matrix multiplication is not commutative. This means that for most matrices, \( A B eq B A \). However, when matrices are symmetric and they commute, the product keeps some of their original properties.

• For symmetric matrices, commutativity can help retain symmetry in products, as shown by \( (A B)^T = A B \) if \( A B = B A \).
• This concept plays a key role in exploring other operations like finding an inverse or taking a transpose.

Inverse Matrix

An inverse matrix is a matrix that, when multiplied by its original matrix, yields the identity matrix. If a matrix \( A \) has an inverse, it is denoted as \( A^{-1} \), and must satisfy \( A A^{-1} = A^{-1} A = I \), where \( I \) is the identity matrix and acts as the neutral element in matrix multiplication. Only square matrices—those having the same number of rows and columns—can have inverses, and not all square matrices are invertible. The matrix must be non-singular (having a non-zero determinant) for its inverse to exist.

• A practical application of inverse matrices is in solving linear equations, where an equation \( A\mathbf{x} = \mathbf{b} \) can be solved by finding \( \mathbf{x} = A^{-1}\mathbf{b} \).
• In problems involving symmetric matrices, knowing whether an inverse exists can affect properties such as symmetry retention in matrix products involving the inverse.

Matrix Transpose

The transpose of a matrix is essentially the matrix flipped over its diagonal. This means that the rows of the original matrix become the columns in the transposed matrix and vice versa. If a matrix \( A \) is transposed, it is denoted as \( A^T \). The operation of transposing is simple yet powerful and has some important properties:

• The transpose of a transpose brings you back to your original matrix: \((A^T)^T = A \).
• Transposing a product changes the order and transposes each component: \((AB)^T = B^T A^T \).

For symmetric matrices, transpose is especially interesting because a symmetric matrix is defined by the fact that \( A = A^T \). Thus, the operation of transposing a symmetric matrix leaves it unchanged. This characteristic is immensely helpful for computations and can simplify solving many problems, especially when dealing with complex mathematical operations involving symmetric matrices.

Skew-Symmetric Matrix

A skew-symmetric matrix is quite different from a symmetric one. It is a matrix \( A \) that satisfies the condition \( A^T = -A \). Here's what this property means:

• All elements on the main diagonal of a skew-symmetric matrix are zero.
• For all off-diagonal elements, they are the negatives of their "mirror" elements across the diagonal.

For example, a skew-symmetric matrix looks like this: \[ \begin{bmatrix} 0 & -a & -b \ a & 0 & -c \ b & c & 0 \ \end{bmatrix} \] Interestingly, the sum or product of two skew-symmetric matrices is not necessarily skew-symmetric. The concept of skew-symmetry is essential when exploring other types of symmetries and their relationships, such as how they intersect with symmetric or orthogonal matrices.